Celiac Disease, CD or Gluten Enteropathy, is not a food allergy; it is an autoimmune disease.
A true allergy is an immune response, which causes the body to produce a protein called IgE.  The production of IgE causes an immediate reaction in the body.  In the case of a wheat allergy, the reactions to the IgE could be, runny rose, watery eyes, or a rash like hives. The allergy might produce wheezing /asthma symptoms or intestinal distress including, stomachache, nausea, or bloating; less common would be diarrhea (but not unheard of).
Typically with an allergy those symptoms could occur within a half an hour of consuming wheat or gluten. Sometimes people could have a delayed reaction (6 to 12) hours after eating gluten.
Not so with someone with CD or Gluten Enteropathy, this person may have no discernible imediate response to consuming gluten, yet the affect upon their body is far more disruptive.

In a person with CD, the abnormal immune response to gluten is based on their body producing IgG or IgM, when they consume, or, in extreme cases, simply come in contact with gluten.  The production of the proteins IgG or IgM, causes an inflammation of the intestinal tract that gets worse over time.   Often these people will experience intestinal symptoms, but these do not flare up in response to what is eaten.  Instead, over time, the inflammation in the intestines increases; there might be an intensifying of stomach pain, diarrhea, headaches etc.  Because this condition leads to malabsorption of nutrients though the small intestine, people with CD often feel sick, or run down, and these feelings get progressively worse with time.
For people with CD or Gluten Enteropathy, the only treatment is to completely eliminate gluten from their diets.
For someone with a wheat or gluten allergy, they must also eliminate these products from their diet.  Some people overcome or “grow out of” allergies if they avoid the allergen consciously.  In the case of an allergy, it is possible to reinstate small amounts of the irritants after years of abstinence.
For people with CD, they must remain gluten free as a way of life, and if they do, they can look forward to relatively normal lives.
A good discussion of this topic is given by Dr. Michael Marcus in the video below.  You have to put up with the commercial that comes on before his discussion, but it is worth it.

http://www.ask.com/health/hlvideo-5min/comparing-wheat-allergy-and-celiac-disease-306538469